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The general regression model with n observations and k explanators, the first of which is a constant unit vector whose coefficient is the regression intercept, is = + where y is an n × 1 vector of dependent variable observations, each column of the n × k matrix X is a vector of observations on one of the k explanators, is a k × 1 vector of true coefficients, and e is an n× 1 vector of the ...
It is calculated as the sum of squares of the prediction residuals for those observations. [ 1 ] [ 2 ] [ 3 ] Specifically, the PRESS statistic is an exhaustive form of cross-validation , as it tests all the possible ways that the original data can be divided into a training and a validation set.
If that sum of squares is divided by n, the number of observations, the result is the mean of the squared residuals. Since this is a biased estimate of the variance of the unobserved errors, the bias is removed by dividing the sum of the squared residuals by df = n − p − 1, instead of n , where df is the number of degrees of freedom ( n ...
In statistics, deviance is a goodness-of-fit statistic for a statistical model; it is often used for statistical hypothesis testing.It is a generalization of the idea of using the sum of squares of residuals (SSR) in ordinary least squares to cases where model-fitting is achieved by maximum likelihood.
Mathematically, the definition of the residual for the i th observation in the data set is written = (; ^), An illustrative plot of a fit to data (green curve in top panel, data in red) plus a plot of residuals: red points in bottom plot.
Using matrix notation, the sum of squared residuals is given by S ( β ) = ( y − X β ) T ( y − X β ) . {\displaystyle S(\beta )=(y-X\beta )^{T}(y-X\beta ).} Since this is a quadratic expression, the vector which gives the global minimum may be found via matrix calculus by differentiating with respect to the vector β {\displaystyle \beta ...
In ordinary least squares, the definition simplifies to: =, =, where the numerator is the residual sum of squares (RSS). When the fit is just an ordinary mean, then χ ν 2 {\displaystyle \chi _{\nu }^{2}} equals the sample variance , the squared sample standard deviation .
Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals.